What is i^i

i^{i } = 0.207879576...
i = \sqrt{-1}

If a is algebraic and b is algebraic but irrational then a^b is transcendental. (Gelfond-Schneider Theorem)

Since i is algebraic but irrational, the theorem applies.

1. We know
e^{ix}= \cos x + i \sin x

Let x = \pi/2

2. e^{i \pi/2} = \cos \pi/2 + i \sin \pi/2

\cos \pi/2 = \cos 90^\circ = 0

\sin 90^\circ = 1
i \sin 90^\circ = (i)*(1) = i

3. Therefore
e^{i\pi/2} = i
4. Take the ith power of both sides, the right side being i^i and the left side =
(e^{i\pi/2})^{i}= e^{-\pi/2}
5. Therefore
i^{i} = e^{-\pi/2} = .207879576...

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